Arectangular area is to be enclosed and divided into thirds. the family has $1040 to spend for the fencing material. the outside fence costs $10 per foot installed, and the dividers cost $20 per foot installed. what are the dimensions that will maximize the area enclosed? (the answer contains a fraction.)

  26 m long by 8 2/3 m wide (dividers are 8 2/3 m)

Step-by-step explanation:

Let x represent the length of the side of the rectangular area parallel to the partitions. Then the cost of the ends and partitions is ...

  10x +20x +20x +10x = 60x

The cost of the other two sides of the rectangle will be (1040 -60x), so those sides will have length ...

  (1/2)(1040 -60x)/10 = 52 -3x

The area of the rectangle is the product of its two dimensions ...

  area = x(52 -3x)

This is a downward-opening parabola that has zeros at x=0 and x=52/3. The vertex of the parabola will be halfway between these values, at ...

  x = (0 +52/3)/2 = 26/3

  52 -3(26/3) = 26 . . . . other side dimension

The area is maximized when the enclosure side parallel to the partitions is 8 2/3 meters, and the other side is 26 meters.

Comment on this problem

By making the cost of the partitions double, the problem effectively becomes one of maximizing the areas of three separate enclosures. The area of each of those is maximum when each is a square, $1040/(3 enclosures×4 sides/enclosure×$10/m) = 8 2/3 m/side. When these are placed side-by-side, the rectangle is 8 2/3 m by 3×(8 2/3 m), or 8 2/3 m by 26 m.

Comment on this type of problem

This sort of problem has the solution that the area of the rectangle is maximized for the cost when half the cost is spent in each of the orthogonal directions of the sides of the rectangle. This is true for a variety of geometries, including ones where all or part of a side or partition may be missing.

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is